A fractional model of viscoelastic relaxation

Materials Science and Engineering A 370 (2004) 209–212

A fractional model of viscoelastic relaxation G. Haneczok a,∗ , M. Weller b a

Institute of Physics and Chemistry of Metals, Silesian University, Bankowa 12, 40-007 Katowice, Poland b Max-Planck Institut für Metallforschung, Heisenbergstr 3, 70569 Stuttgart, Germany Received 12 July 2002

Abstract The fractional model of viscoelastic relaxation was applied for the analysis of the high temperature internal friction background in two materials: a ␥-TiAl-based alloy and a tetragonal polycrystalline zirconia ceramic (ZrO2 –2.8 mol% Y2 O3 ). In both cases, the fractional model offers a consistent description of the experimental data showing that the high temperature internal friction background exhibits viscoelastic behaviour. The fractional parameter q increases with increasing temperature, which reflects the increasing role of a Newtonian fluid component in the viscoelastic relaxation. © 2003 Elsevier B.V. All rights reserved. Keywords: Internal friction; Viscoelastic relaxation; ␥-TiAl alloys; Tetragonal zirconia polycrystals

1. Introduction Recently, in a series of papers [1–7], it was shown that high temperature internal friction background in various materials is of viscoelastic nature. In ␥-TiAl-based alloys [1–3], NiAl single crystals [4], polycrystalline cubic and tetragonal zirconia ceramics [5–7] it was observed that the magnitude of damping at constant temperature is proportional to ω−n (where ω is the measuring frequency and n = 0.2–0.5). The analysis of the high temperature internal friction background is essentially based on the Maxwell rheological model representing ideal viscoelastic behaviour [8,9]. In order to improve our understanding of the high temperature background, we consider the fractional model of viscoelasticity originally formulated in [10,11] by a straightforward application of the fractional calculus to the Maxwell unit. Our goal in the present paper is to test the applicability of this model to high temperature internal friction background analysis. For such test, we use experimental data of two materials: (i) A ␥-TiAl-based two phase alloy with fine grained primary annealed microstructure [1]; and (ii) a tetragonal polycrystalline ZrO2 –2.8 mol% Y2 O3 [6,7]. For both materials, high temperature internal friction back-

ground was measured in a wide frequency range from 10−2 to 10 Hz up to sufficiently high temperatures, i.e. 1300 and 1600 K, respectively. The results are compared with that reported in [1,6,7].

2. Fractional model The fractional model is based on the application of fractional calculus to the original Maxwell unit, which is a mechanical analogue to a viscoelastic body and consists of a spring and a dashpot in series. In this rheological model, the spring represents linear elasticity of a solid for which the stress σ and the strain ε are coupled by the Hooke relation σ = Mε, where M is the modulus. A viscous flowing of material, or a Newtonian fluid, is represented by a dashpot with a viscosity η for which σ = η(dε/dt). In the case of a linear viscous fluid, η is taken to be independent on stress. This assumption allows introducing a time constant τ representing the relaxation time with η = τM. For the Maxwell unit (ideal viscoelasticity), the constitutive differential equation is [8,9]: τ

∗

Corresponding author. E-mail address: [email protected] (G. Haneczok).

0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.01.009

dσ dε + σ = τM dt dt

(1)

For dynamic experiments with periodic stress and strain σ = σ0 exp[(iωt)],  = 0 exp[i(ωt − φ)](φ = loss angle) the

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internal friction Q−1 = tanφ is obtained from Eq. (1) as 1 Q−1 = (2) ωτ Experimental data analysis shows that the original Maxwell unit, representing ideal viscoelasticity, is not able to describe a real viscoelastic body. Therefore, the original Maxwell model was empirically generalised as 1 (3) Q−1 = (ωτ)n In this generalised Maxwell model, deviation from ideal viscoelasticity, which corresponds to n = 1, can be considered as being due to a distribution of relaxation times [12]. The fractional model of viscoelastic relaxation can be introduced by a formal replacing of the first-order time derivative d/dt in Eq. (1) by a fractional derivative dq /dtq of non-integer order q(0 < q ≤ 1), so we have: dq ε dq σ (4) τ q + σ = τM q dt dt From a physical point of view, this means that viscoelasticity is described via an internal parameter q characterising a deviation from a pure elasticity and/or a pure flowing. Indeed, viscoelastic bodies are neither a Hookean solid nor a Newtonian fluid but something in between. For an ideal Newtonian fluid, we have σ = η dε/dt with q = 1 and for an ideal Hookean solid σ ∝ ε, which means q = 0. The intermediate case corresponds to σ ∝ dq ε/dt q with 0 < q ≤ 1. The solution of the fractional differential Eq. (4) with σ = σ0 exp(iωt), as shown by Nonnenmacher et al. [10,11], leads to the following expression for the internal friction Q−1 (ω, τ, q): sin(πq/2) (5) Q−1 (ω, τ, q) = cos(πq/2) + (ωτ)q For q = 1, Eq. (5) gives the solution of the standard Maxwell unit, i.e. Q−1 (ω, τ) = (ωτ)−1 . Since viscoelastic behaviour is determined by thermally activated processes an Arrhenius equation holds for the relaxation time τ.
 H τ −1 = τ∞ −1 exp − (6) kB T −1 is the attempt frequency and H the activation where τ∞ enthalpy (kB is the Boltzmann constant). Application of Eq. (5) to the experimental data consists in determination of q and τ as a function of temperature. The typical experimental data consist of Q−1 versus temperature curves measured for different constant frequencies (see [1]). From these data, one can determine Q−1 (ω) for different T = constant. From Eq. (5) we have:

ln[Y(T, ω, q)] = qln(ω) + qln[τ(T)]

(7)

where Y(T, ω, q) = [sin(πq/2)/Q−1 (T, ω)] − cos(πq/2). With the evident assumption that ωτ > 0, Q−1 > 0 and that Q−1 (ω) is a continuously decreasing function, it can be shown that the non-linear Eq. (7) has only one unique solution which can be obtained by applying a standard iteration

procedure. In the first attempt, the left side of Eq. (7) for T = constant and q = q0 = 1, which is ln[Y(T, ω, 1)] = −ln(Q−1 (T = constant, ω), is plotted versus ln(ω). The values of q and ln(τ) can be calculated as the parameters of the expected straight line (q = q1 = slope and (lnτ)1 = intercept/q1 ). In the second attempt, the estimated value of q = q1 is used to calculate the function Y(T,ω,q1 ), which allows again to plot the left side of Eq. (7) versus ln(ω). The iteration procedure is then defined as: ln[Y(T, ω, qi )] = qi+1 ln(ω) + qi+1 ln[τi+1 (T)]

(8)

with i = 0 · · · m, and q0 = 1 (m is the number of iterations leading to qm ≈ qm+1 ). In typical cases, a stable solution with a precision of at least three significant digits can be reached after several iterations. 3. Comparison with experiment Measurements of the following samples were evaluated: (i) ␥-TiAl-based alloy with a nominal composition of Ti–46.5 at.% Al–4 at.% (Cr, Nb, Ta, B) with a fine grained microstructure (for details see [1–3,13]); (ii) tetragonal zirconia polycrystals (ZrO2 –2.8 mol% Y2 O3 ) denoted as 3Y-TZP [6,7]. Mechanical loss experiments were carried out with a sub-resonance torsion apparatus using forced vibrations [14]. Mechanical loss represented by Q−1 ≡tanφ was determined with increasing temperature for various frequencies between 10−2 and 10 Hz in the temperature range from 300 to 1280 K for ␥-TiAl-based alloy and 300 to 1580 K for 3Y-TZP. Fig. 1a and b show mechanical loss spectra, Q−1 (T), obtained for the ␥-TiAl-based alloy and for 3Y-TZP ceramic, respectively. As it can be recognised from these figures the Q−1 versus T curves exhibit two features: (i) above a certain temperature (900 K for ␥-TiAl and 1300 K for 3Y-TZP) the mechanical loss, Q−1 (T), strongly increases with temperature; and (ii) at constant temperature Q−1 increases with decreasing frequency. Both properties are characteristic for viscoelastic behaviour [8,9]. Application of the fractional model to the experimental data is presented in Fig. 2(a) and (b) where the plot of Y(T,ω,q) versus ln(ω) (see Eq. (8)) is shown for several constant temperatures for the ␥-TiAl and the 3Y-TZP, respectively. For both examined materials, the solution of Eq. (8), i.e. qm is found to be temperature dependent. This is demonstrated in Fig. 3 where qm is plotted versus T. Fig. 4 shows the corresponding Arrhenius plot from which the activation −1 can be deenthalpies H and the pre-exponential factors τ∞ duced as: −1 = 5.9× 1013 s−1 γ − TiAl : H = (3.63 ± 0.06) eV, τ∞

and −1 3Y -TZP : H = (6.38 ± 0.02) eV, τ∞ = 6.4 ×1018 s−1 .

G. Haneczok, M. Weller / Materials Science and Engineering A 370 (2004) 209–212

0.5

4 0.01 Hz

0.4

211

γ -Ti-46.5at% Al

3

γ -Ti-46.5at% Al

-4at% (Cr,Nb,Ta,B)

-4at% (Cr,Nb,Ta,B) 0.04 Hz

2 ln[Y(q)]

0.3 Q

-1

0.10 Hz

0.2

0.0 900

0

0.40 Hz 1.00 Hz 4.00 Hz 10.0 Hz

0.1

1

-1

HT86

1000

(a)

1100 1200 T [K]

1300

1400

-4 -3 -2 -1 0

Q

-1

0.10

0.05

0.00 1200 (b)

2

3

4

5

3Y-TZP 1425 K qm=0.32 1475 K qm=0.33 1525 K qm=0.34

4 ln[Y(q)]

0.15

1

ln( ω [Hz])

(a)

5 3Y-TZP 0.01 Hz 0.04 Hz 0.10 Hz 0.40 Hz 1.00 Hz 4.00 Hz 10.0 Hz

1150 K, qm=0.31 1200 K, qm=0.34 1250 K, qm=0.37

3 2

1300

1400

1500

1

1600

T [K]

Q−1

Fig. 1. (a) vs. T for different frequencies for ␥-TiAl-based alloy; (b) Q−1 vs. T for different frequencies for 3Y-TZP ceramic.

4. Discussion The plot of ln(Y(T,ω,q)) versus ln(ω) in Fig. 2(a) and (b) show a good linear correlation indicating that the fractional model can be applied to describing the experimental data.

-3

-2

-1

0

1

2

3

4

5

ln( ω [Hz])

(b)

Fig. 2. (a) Plot of ln(Y(T,ω,q)) vs. ln(ω) according to Eq. (8) for ␥-TiAl-based alloy; (b) Plot of ln(Y(T,ω,q)) vs. ln(ω) according to Eq. (8) for 3Y-TZP ceramic.

to 0.42 at 1300 K. This dependence reflects the evolution of viscous flow with temperature and indicates that the component of Newtonian fluid in viscoelastic phenomena increases with temperature.

4.1. γ-TiAl 0.35

0.44 γ -TiAl

3Y-TZP

0.40

0.34 qm

−1 = The Arrhenius parameters (H = 3.63 eV and τ∞ 13 −1 5.9 × 10 s ) do not differ too much from those, which we obtained with the generalised Maxwell model (Eq. (3)) in −1 = 2.3 × 1013 s−1 ). The lower ac[1] (H = 3.9 eV and τ∞ tivation enthalpy obtained for the fractional model is closer to that deduced from the creep experiments (3.5–3.7 eV [1]) and to the H value for Al self-diffusion in TiAl-based alloys (3.7 eV [15]). These facts confirm earlier conclusions formulated in [1,2], i.e. the high temperature internal friction background exhibits viscoelastic behaviour. The main difference between the results obtained by applying the fractional model and the generalised Maxwell unit for ␥-TiAl-based alloy, consists in determining the temperature dependence of the fractional parameter q (see Fig. 3). With increasing temperature q increases from 0.30 at 1100 K

0.36 0.33 0.32 0.32 1100 1200 1300 1400 1500 1600 T [K]

Fig. 3. Fractional parameters q vs. temperature for ␥-TiAl alloy and 3Y-TZP ceramic.

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10

ln(τ )

8 6

3Y-TZP

H=(6.38±0.02) eV -1 18 -1 τ∞ =6.4 10 s

γ-TiAl

H=(3.63±0.06) eV -1 13 -1 τ∞ =5.9 10 s

4

ramic leads to the following conclusions: (i) the fractional model of viscoelastic relaxation offers a consistent description of the high temperature internal friction background; (ii) The obtained activation enthalpies are very close to those deduced by applying the generalised Maxwell model; (iii) for both examined materials, the fractional parameter q increases with increasing temperature, which reflects the increasing role of a Newtonian fluid component in the viscoelastic phenomena.

2

References 6.5

7.0

7.5

8.0

8.5

9.0

-1

10000/T [K ] Fig. 4. Arrhenius plot for ␥-TiAl-based alloy and 3Y-TZP ceramic.

4.2. Tetragonal zirconia ceramics The activation enthalpy (H = 6.38 eV) also agrees with that reported in [6,7] (H = 6.5 eV), which was determined by applying the generalised Maxwell model. The results presented in Fig. 3 show that the fractional parameter q increases with temperature. So, again it can be stated that the Newtonian fluid component in viscoelastic phenomena increases with temperature. Apparently, this increase is smaller in 3Y-TZP compared with ␥-TiAl, which can be explained by the higher melting point of the ceramic material. 5. Conclusions Application of the fractional model of viscoelastic relaxation for the analysis of the high temperature internal friction background in the ␥-TiAl-based alloy and 3Y-TZP ce-

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